we think of each such presheaf as being a rule that assigns to each test space U∈CU \in C the set X(U)X(U) of allowed maps from UUinto the would-be space XX (this is really the perspective of functorial geometry, originally due to Grothendieck 65);

we think of each such copresheaf AA as a rule that assigns to each test space U∈CU \in C the set A(U)A(U) of allowed maps from the would-be space AA into UU, hence as the collection of UU-valued functions on AA. Since a function on a point is a “quantity”, these are generalized quantities.

One may view the Yoneda lemma and the resulting Yoneda embedding as expressing consistency conditions on this perspective: The Yoneda lemma says that the prescribed rule for how to test a generalized space XX by a test space UU turns out to coincide with the actual maps from UU to XX, when UU is itself regarded as a generalized space, and the Yoneda embedding says that, as a result, the nature of maps between test spaces does not depend on whether we regard these as test spaces or as generalized spaces.

Beyond this automatic consistency condition, guaranteed by category theory itself, typically the admissible (co)presheaves that are regarded as generalized spaces and quantities are required to respect one more consistency condition:

If CC carries the structure of a site, one asks a generalized space to be a presheaf X=PSh(C)=[Cop,Set]X = PSh(C) = [C^{op},Set] that respects the way objects in CC are covered by other objects. These are the sheaves. The category of sheaves

Given any generalized spaces, functions out of it are expected to respect products of coefficient objects, in that a function with values in U×VU \times V is the same as a pair of functions, one with values in UU, one with values in VV. Hence one is typically interested in copresheaves that preserve at least product

Details

which underlies much of mathematics is at its heart controlled by the following elementary category theoretic reasoning:

Let SS be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance S=ΔS = \Delta, the simplicial category, or S=S = CartSp.

An ordinary manifold, for instance, is a space required to be locally isomorphic to an object in S=CartSpS = CartSp. But more generally, a space XX modeled on SS need only be probeable by objects of SS, giving a rule which to each test object U∈SU \in S assigns the collection of admissible maps from UU to XX, such that this assignment is well-behaved with respect to morphisms in SS. Such an assignment is nothing but a presheaf on SS, i.e. a contravariant functor

X:Sop→Set.
X : S^{op} \to Set
\,.

Therefore general spaces modeled on SS are nothing but presheaves on SS:

SpacesS:=PSh(S).
Spaces_S := PSh(S)
\,.

Of course this is an extremely general notion of spaces modeled on SS.

We have the Yoneda embeddingS↪SpacesSS \hookrightarrow Spaces_S and using this we can say that the collection of functions on a generalized space XX with values in U∈SU \in S is

C(X,U):=HomSpacesS(X,U).
C(X,U) := Hom_{Spaces_S}(X,U)
\,.

This assignment is manifestly covariant in UU, and hence more generally we can consider the functions on XX, C(X)C(X) to be a copresheaf on SS, namely a covariant functor

C(X):=Hom(X,−):S→Sets.
C(X) := Hom(X,-) : S \to Sets
\,.

One can think of C(X)C(X) as being a generalized quantity which may be co-probed by objects of SS.

In this vein, one can say, generally, that co-presheaves on SS are generalized quantities modeled on SS, and we write

QuantitiesS:=CoPSh(S).
Quantities_S := CoPSh(S)
\,.

Given any such generalized quantity A∈QuantitiesSA \in Quantities_S, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called Spec(A)Spec(A) and can be defined as a presheaf by the assignment

In conclusion, the grand duality between spaces and quantities is a consequence of the formal duality which reverses the arrows in the category SS of test spaces.

This story generalizes straightforwardly from presheaves with values in Set to presheaves with values in other categories. Of relevance are in particular presheaves with values in the category Top of topological spaces and presheaves with values in the category of spectra. See the examples below.

where we used the Yoneda lemma[Cop,V](j(v),j(u))≃V(v,u)[C^{op},V](j(v),j(u)) \simeq V(v,u) and the co-Yoneda lemmaX≃∫v∈Vj(v)⋅X(v)X \simeq \int^{v \in V} j(v) \cdot X(v) and the fact that the VV-enriched hom sends colimits and coends in the first argument to limits and ends.